The theory of sharp first-order univalence criteria
Let \(D \subset \mathbb{C}\) be a Jordan domain such that the derivative of the Riemann mapping function from the unit disk \(\mathbb{D}\) onto \(D\) has a continuous and nonvanishing extension to \(\overline \mathbb{D}\) \((\partial D\) Dini-smooth e.g.). The case \(D = H\) where \(H\) is the upper half-plane also occurs as a certain limit case. Another domain \(R \subset \mathbb{C}\) is now called a domain of univalence for \(D\) if, for every function \(f\) analytic in \(D\), the condition \(f'(D) \subset R\) implies that \(f\) is univalent. A well-known simple example is the Noshiro-Warschawski univalence criterion where \(D\) is any convex domain and \(R\) is some halfplane \(e^{i \theta}H\). Another interesting example is obtained if \(D = \mathbb{D}\) and \(R = \{1 < |w |< \gamma\}\), where \(\gamma \geq e^{\pi/2}\) is the (unknown) John constant. Generalizing the last example the author considers doubly connected domains of univalence of the type \(R = S_1 \backslash \overline {S_2}\) where \(S_1\), \(S_2\) \((\overline {S_2} \subset S_1)\) are strictly starshaped domains with analytic boundary curves. The univalence criterion \(f'(D) \subset R\) is now called (strongly) sharp if, for every domain \(R' \subsetneqq R\) there exists a nonunivalent function \(f\) with \(f'(D) \subset R'\). It is shown that this sharpness condition is closely related to the existence of some extremal function \(f_0\) still satisfying \(f_0'(D) \subset R\) but such that there are points \(a,b \in \partial D\) with \(f_0(a) = f_0(b)\). In order to characterize such extremal functions an interesting variational method is developed from which it follows as a basic result that \(f_0'(D)\) covers the universal covering surface of \(R\) a finite number of times. More precisely it is shown that \(f_0'(z) = \Gamma (k(z))\) where \(k : D \to S = \{0 < \text{Re} \zeta < 1\}\) is analytic such that \(\text{Re} k\) is the harmonic measure of a finite number of arcs on \(\partial D\) and \(\Gamma\) maps the strip \(S\) \(1 - 1\) onto the universal covering surface of \(R\). The author shows, in addition, that the determination of a large class of sharp first order criteria of the type described above can be reduced to the solution of certain extremal problems involving a finite number of parameters.
scientific article; zbMATH DE number 718519
scientific article